Equational Axioms Research Articles

A note on identities of two-dimensional languages

In this note we consider identical laws satisfied by two-dimensional (picture) languages, collections of rectangular arrays over a given alphabet. We prove that an identity α = β holds for all picture languages if and only if α and β represent the same bi-language (a subset of a free bi-monoid). As a consequence, we obtain decidability of the equational theory of picture languages, a description of free objects in the variety generated by picture language algebras, and prove that such a variety does not have a finite equational axiomatization.

Bisimilarity is not Finitely Based over BPA with Interrupt

This paper shows that bisimulation equivalence does not afford a finite equational axiomatization over the language obtained by enriching Bergstra and Klop's Basic Process Algebra with the interrupt operator. Moreover, it is shown that the collection of closed equations over this language is also not finitely based.

Nested Semantics over Finite Trees are Equationally Hard: Invited Talk

One of the criteria that has been put forward for studying the mathematical tractability of the behavioural equivalences in van Glabbeek's linear time- banching time spectrum is that they afford elegant, finite equational axiomatizations over fragments of process algebraic languages. A review of existing complete equational axiomatizations for many of the behavioural semantics in van Glabbeek's spectrum is offered in his encyclopedic chapter in the [Handbook of Process Algebra, Jan A. Bergstra, Alban Ponse, and Scott A. Smolka, Editors. North-Holland, Amsterdam, 2001]. The equational axiomatizations offered ibidem are over the language BCCSP, a common fragment of Milner's CCS and Hoare's CSP suitable for describing finite synchronization trees, and characterize the differences between behavioural semantics in terms of a few revealing axioms.The main omissions in this menagerie of equational axiomatizations for the behavioural semantics in van Glabbeek's spectrum are axiomatizations for 2-nested simulation semantics and possible futures semantics. The relation of 2-nested simulation was introduced by Groote and Vaandrager as the coarsest equivalence included in completed trace equivalence for which the tyft/tyxt format is a congruence format. Possible futures semantics, on the other hand, was proposed by Brookes and Rounds as far back as 1981, and is one of the nested trace equivalences introduced by Hennessy and Milner.In this talk, I shall present, amongst other results, a mathematical answer to the following natural question:Why have these semantics not been given complete axiomatizations yet, even for the language of finite synchronization trees?More precisely, I shall argue that the 2-nested simulation and the possible futures preorder and equivalence do not admit a finite (in)equational axiomatization over the language BCCSP. I then generalize these negative results to show that, for n⩾2, none of the n-nested simulation or trace preorders and equivalences defined by Groote and Vaandrager, and Hennessy and Milner, respectively, afford finite equational axiomatizations over the language BCCSP.Interestingly, the intersection of all of the n-nested simulation or trace equivalences or preorders over image-finite labelled transition systems, and therefore over the language BCCSP, is bisimulation equivalence. Hennessy and Milner proved that bisimulation equivalence is axiomatized over the language BCCSP by means of four classic equations. It follows that this fundamental behavioural equivalence, albeit finitely based over BCCSP, is the limit of sequences of relations that do not afford finite equational axiomatizations themselves. Thus “finite axiomatizability” is an example from process theory of a “discontinuous” property of a behavioural equivalence—i.e., of a property that “appears at the limit”, but is not afforded by its finite approximations. This talk is based upon joint work with Wan Fokkink (CWI), Rob van Glabbeek (INRIA, Sophia Antipolis) and Anna Ingólfsdóttir (BRICS, Aalborg University).

Nested semantics over finite trees are equationally hard

This paper studies nested simulation and nested trace semantics over the language BCCSP, a basic formalism to express finite process behaviour. It is shown that none of these semantics affords finite (in)equational axiomatizations over BCCSP. In particular, for each of the nested semantics studied in this paper, the collection of sound, closed (in)equations over a singleton action set is not finitely based.

Split-2 Bisimilarity has a Finite Axiomatization over CCS with Hennessy's Merge

This note shows that split-2 bisimulation equivalence (also known as timed equivalence) affords a finite equational axiomatization over the process algebra obtained by adding an auxiliary operation proposed by Hennessy in 1981 to the recursion free fragment of Milner's Calculus of Communicating Systems. Thus the addition of a single binary operation, viz. Hennessy's merge, is sufficient for the finite equational axiomatization of parallel composition modulo this non-interleaving equivalence. This result is in sharp contrast to a theorem previously obtained by the same authors to the effect that the same language is not finitely based modulo bisimulation equivalence.

CCS with Hennessy's Merge has no Finite Equational Axiomatization

This paper confirms a conjecture of Bergstra and Klop's from 1984 by establishing that the process algebra obtained by adding an auxiliary operator proposed by Hennessy in 1981 to the recursion free fragment of Milner's Calculus of Communicationg Systems is not finitely based modulo bisimulation equivalence. Thus Hennessy's merge cannot replace the left merge and communication merge operators proposed by Bergstra and Klop, at least if a finite axiomatization of parallel composition is desired.

Nested Semantics over Finite Trees are Equationally Hard

This paper studies nested simulation and nested trace semantics over the language BCCSP, a basic formalism to express finite process behaviour. It is shown that none of these semantics affords finite (in)equational axiomatizations over BCCSP. In particular, for each of the nested semantics studied in this paper, the collection of sound, closed (in)equations over a singleton action set is not finitely based.

Squares in Fork Arrow Logic

In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is U×U for some set U) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Nemeti, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares.

On the equational definition of the least prefixed point

We propose a method to axiomatize by equations the least prefixed point of an order preserving function. We discuss its domain of application and show that the Boolean modal μ-calculus has a complete equational axiomatization. The method relies on the existence of a “closed structure” and its relationship to the equational axiomatization of Action Logic is made explicit. The implication operation of a closed structure is not monotonic in one of its variables; we show that the existence of such a term that does not preserve the order is an essential condition for defining by equations the least prefixed point. We stress the interplay between closed structures and fixed point operators by showing that the theory of Boolean modal μ-algebras is not a conservative extension of the theory of modal μ-algebras. The latter is shown to lack the finite model property.

Theorem Proving Modulo

Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of general interest because it permits one to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof-theoretic account of the combination of computations and deductions. The congruence on propositions is handled through rewrite rules and equational axioms. Rewrite rules apply to terms but also directly to atomic propositions. The second contribution is to give a complete proof search method, called extended narrowing and resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higher-order logic can be presented as a theory in deduction modulo. Applying the ENAR method to this presentation of higher-order logic subsumes full higher-order resolution.

The multiplicative fragment of the Yanov equational theory

We give a finite equational axiomatization for +-free identities of (regular) languages which contain the empty word. The axioms for the whole equational theory of such languages were given by Yanov.

Greibach Normal Form in Algebraically Complete Semirings

We give inequational and equational axioms for semirings with a fixed-point operator and formally develop a fragment of the theory of context-free languages. In particular, we show that Greibach's normal form theorem depends only on a few equational properties of least pre-fixed-points in semirings, and elimination of chain- and deletion rules depend on their inequational properties (and the idempotency of addition). It follows that these normal form theorems also hold in non-continuous semirings having enough fixed-points.

On axioms for commutative regular equations without addition

Algebras of commutative languages consist of all subsets of a free commutative monoid over a given alphabet Σ, and they are endowed with the operations of union, complex multiplication, Kleene iteration (submonoid generation), and the empty set and the set whose unique element is the unit of the free monoid, as constants. There is a well-known equational axiomatization for these algebras. However, any such axiomatization is necessarily infinite. Now, by removing the operation of union (addition) from the described algebras, the corresponding equational theory reduces to the collection of those equations which contain no symbols of addition. We supply a nontrivial list of equational axioms for the so obtained equational theory, and prove that it has no finite equational base, too.

An Equational Axiomatization of Bisimulation over Regular Expressions

We provide a finite equational axiomatization for bisimulation equivalence of nondeterministic interpretation of regular expressions. Our axiomatization is heavily based on the one by Salomaa, that provided an implicative axiomatization for a large subset of regular expressions, n amely all those that satisfy the non-empty word property (i.e. without 1 summands at the top level) in -contexts. Our restriction is similar, it essentially amounts to recursively requiring that the non-empty word property be satisfied not just at top level but at any depth. We also discuss the impact on the axiomatization of different interpretations of the term, interpreted either as a null process or as a deadlock.

Equational Axioms for Probabilistic Bisimilarity (Preliminary Report)

This paper gives an equational axiomatization of probabilistic bisimulation equivalence for a class of finite-state agents previously studied by Stark and Smolka ((2000) <em>Proof, Language, and Interaction: Essays in Honour of Robin Milner</em>, pp. 571-595). The axiomatization is obtained by extending the general axioms of iteration theories (or iteration algebras), which characterize the equational properties of the fixed point operator on (omega-)continuous or monotonic functions, with three axiom schemas that express laws that are specific to probabilistic bisimilarity. Hence probabilistic bisimilarity (over finite-state agents) has an equational axiomatization relative to iteration algebras.

Operators and Laws for Combining Preference Relations

The paper is a theoretical study of a generalization of the lexicographic rule for combining ordering relations. We define the concept of priority operato r: a priority operator maps a family of relations to a single relation which represents their lexicographic combination according to a certain priority on the family of relations. We present four kinds of results. We show that the lexicographic rule is the only way of combining preference relations which satisfies natural conditions (similar to those proposed by Arrow). We show in what circumstances the lexicographic rule propagates various conditions on preference relations, thus extending Grosof’s results. We give necessary and sufficient conditions on the prior ity relation to determine various relationships between combinations of preferences. We give an algebraic treatment of this form of generalized prioritization. Two operators, called but and on the other hand, are sufficient to express any prioritization. We pres ent a complete equational axiomatization of these two operators. These results can be applied in the theory of social choice (a branch of economics), in non-monotonic reasoning (a branch of artificial intelligence), and more generally wherever relations have to be combined.

Short Single Axioms for Boolean Algebra

We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke. Automated deduction techniques were used in several parts of the work.

The algebra of stream processing functions

Data flow networks are a model of concurrent computation. They consist of a collection of concurrent asynchronous processes which communicate by sending data over FIFO channels. In this paper we study the algebraic structure of the data flow networks and base their semantics on stream processing functions. Our algebraic theory is based on the calculus of flownomials. With an additive (or cantorian) interpretation the calculus gives a unified presentation of the classical algebraic models for control structures, that is, regular algebra and iteration theories. The kernel of the calculus is an equational axiomatization called basic network algebra (BNA) for flow graphs modulo graph isomorphism. We show that the algebra of stream processing functions called SPF (used for deterministic networks) and the algebra of sets of stream processing functions called P SPF (used for nondeterministic networks) are BNA algebras. Actually they give a multiplicative (or cartesian) interpretation of the calculus of flownomials. As a byproduct this shows that both semantic models are compositional. This means the semantics of a network may be described in terms of the semantics of its components. (As it is well known this is not true for the input–output relational semantics of nondeterministic networks.) We also identify additional axioms satisfied by the branching constants in these two algebraic theories. For the deterministic case we study in addition the coarser equivalence relation on networks given by the input–output behavior and provide a correct and complete axiomatization.

On Circuit Valuation of Matroids

The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R∪{−∞})-valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of (R∪{−∞})-valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids.

2-Nested Simulation is not Finitely Equationally Axiomatizable

<p>2-nested simulation was introduced by Groote and Vaandrager<br /> [10] as the coarsest equivalence included in completed trace equivalence<br /> for which the tyft/tyxt format is a congruence format. In the linear<br />time-branching time spectrum of van Glabbeek [8], 2-nested simulation<br />is one of the few equivalences for which no finite equational axiomatization<br /> is presented. In this paper we prove that such an axiomatization<br />does not exist for 2-nested simulation.</p><p>Keywords: Concurrency, process algebra, basic CCS, 2-nested simulation,<br /> equational logic, complete axiomatizations.</p>